Theoretical cellular automata have been proposed as models for many physical phenomena from music to quantum mechanics. My question concerns the reverse:

Is there a simple physical system that emulates a theoretical cellular automaton?

A computer running CA software such as Hashlife can be considered an (extremely complex) physical system that behaves according to the rules of the programmed CA. I am however looking for something much simpler, such as a set of dominoes that topple each other in a fashion conforming to the rules of a certain CA, wave interference patterns that model a CA etc.

The point is that there have been enormous advances in CA theory lately (such as the discovery of the first self-replicating pattern in Conway's Game of Life in November 2013) that it would be interesting if these patterns could simply be "set up" in the physical world somehow and then they would evolve according to the CA rules without requiring a computer to simulate the rules. In the long run, this might even lead to the creation self-replicating physical systems.

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I don't know of any system which is modeled by cellular automata themselves, but diffusion limited aggregation is a kissing cousin of cellular automata that does have interesting research implications. In particular, I'll tell you about snow flakes.

Numerical models for snow flake formation use diffusion limited aggregation to grow the flakes at different temperatures with different humidities (see this awesome paper). The reason this is similar to cellular automata is because as the flake grows the presence of other water molecules in the crystal changes the likelihood that nearby portions of the crystal will be able to grow. Specifically, as certain parts grow larger they become more likely to grow while their neighbors become less likely. As a result, the time evolution of any part of the snow crystal is highly dependent upon the evolution of other nearby parts, and the configuration as a whole is very sensitive to initial conditions (both of which are qualities of cellular automata).

The numerical models have also resulted in a better understanding of the intricate designs of snow flakes. For instance the fine, detailed etchings on snow flakes are not etchings at all: they are air bubbles. They are regions that were killed off by the growth of their neighbors and eventually sealed inside the ice.

If you are not interested in the models which the wikipedia article references (biology and speculative physics), I suspect you will find that cellular automata are too simplified and idealized to be used as a physical model whole-cloth.